How to set up the IK Controller bone without separating its parented bone in the process?
This is happening because of where you've set the bone to rotate around. In the video, he has the IK bone and the foot bone rotating in the same spot. In your file, the IK bone is rotating around a point off to the side of the foot bone (at the center of the sphere). This means that any bone that is parented to that IK bone will move in an arc around that IK bone.

Think of it this way: when parenting objects/bones you're telling Blender that bone A has to always be at some position relative to bone B. Let's say that this position is 3 inches to the left of bone B along the X axis of bone B. When we rotate bone B, it's local axes rotate with it. So after rotating bone B, it's X axis could be pointing straight up, or forward, off to one side at 56.67809 degrees, you get my point. But bone A will still follow along, and be 3 inches along the X axis, wherever it points.

So if you move your IK bone to have it's base where the base of your foot bone is, everything should be good to go(try setting the cursor at the base of the foot then using Snap to Cursor). Since the sphere will now be on top of the foot, the foot will be hard to see. If you select the IK bone and go to where you selected the custom shape you can choose wireframe to make it see through.